Biharmonic surfaces in Euclidean space E^3 are fi rstly studied from a diff erential geometric point of view by Bang-Yen Chen, who showed that the only biharmonic surfaces are minimal ones. ,0=A surface x : M^2 ---> E^3 is called biharmonic if Δ^2x -where Δ is the Laplace operator of M^2. We study the L_k- biharmonic spacelike hypersurfaces in the 4-dimentional pseudo- Euclidean space E^4_1 with an additional condition that the principal curvatures of M^3 are distinct. A hypersurface x : M^3 ---> E^4 is called L_k-biharmonic if L_k x = 0 (for k = 0; 1; 2), where Lk is the lin- earized operator associated to the r-th variation of (k+1)-th mean curvature of M^3. Since L0 = Δ, the matter of Lk-biharmonicity is a natural generalization of biharmonicity. On any Lk-biharmonic spacelike hypersurfaces in E41 with distinct principal curvatures, by, assuming Hk to be constant, we get that Hk+1 is constant. Fur- thermore, we show that Lk-biharmonic spacelike hypersurfaces in E41 with constant Hk are k-maximal.
